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Mirrors > Home > HSE Home > Th. List > hlimadd | Structured version Visualization version GIF version |
Description: Limit of the sum of two sequences in a Hilbert vector space. (Contributed by Mario Carneiro, 19-May-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
hlimadd.3 | ⊢ (𝜑 → 𝐹:ℕ⟶ ℋ) |
hlimadd.4 | ⊢ (𝜑 → 𝐺:ℕ⟶ ℋ) |
hlimadd.5 | ⊢ (𝜑 → 𝐹 ⇝𝑣 𝐴) |
hlimadd.6 | ⊢ (𝜑 → 𝐺 ⇝𝑣 𝐵) |
hlimadd.7 | ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛) +ℎ (𝐺‘𝑛))) |
Ref | Expression |
---|---|
hlimadd | ⊢ (𝜑 → 𝐻 ⇝𝑣 (𝐴 +ℎ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hlimadd.3 | . . . . . 6 ⊢ (𝜑 → 𝐹:ℕ⟶ ℋ) | |
2 | 1 | ffvelcdmda 7035 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐹‘𝑛) ∈ ℋ) |
3 | hlimadd.4 | . . . . . 6 ⊢ (𝜑 → 𝐺:ℕ⟶ ℋ) | |
4 | 3 | ffvelcdmda 7035 | . . . . 5 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → (𝐺‘𝑛) ∈ ℋ) |
5 | hvaddcl 29954 | . . . . 5 ⊢ (((𝐹‘𝑛) ∈ ℋ ∧ (𝐺‘𝑛) ∈ ℋ) → ((𝐹‘𝑛) +ℎ (𝐺‘𝑛)) ∈ ℋ) | |
6 | 2, 4, 5 | syl2anc 584 | . . . 4 ⊢ ((𝜑 ∧ 𝑛 ∈ ℕ) → ((𝐹‘𝑛) +ℎ (𝐺‘𝑛)) ∈ ℋ) |
7 | hlimadd.7 | . . . 4 ⊢ 𝐻 = (𝑛 ∈ ℕ ↦ ((𝐹‘𝑛) +ℎ (𝐺‘𝑛))) | |
8 | 6, 7 | fmptd 7062 | . . 3 ⊢ (𝜑 → 𝐻:ℕ⟶ ℋ) |
9 | ax-hilex 29941 | . . . 4 ⊢ ℋ ∈ V | |
10 | nnex 12159 | . . . 4 ⊢ ℕ ∈ V | |
11 | 9, 10 | elmap 8809 | . . 3 ⊢ (𝐻 ∈ ( ℋ ↑m ℕ) ↔ 𝐻:ℕ⟶ ℋ) |
12 | 8, 11 | sylibr 233 | . 2 ⊢ (𝜑 → 𝐻 ∈ ( ℋ ↑m ℕ)) |
13 | nnuz 12806 | . . 3 ⊢ ℕ = (ℤ≥‘1) | |
14 | 1zzd 12534 | . . 3 ⊢ (𝜑 → 1 ∈ ℤ) | |
15 | eqid 2736 | . . . . 5 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 = 〈〈 +ℎ , ·ℎ 〉, normℎ〉 | |
16 | eqid 2736 | . . . . . 6 ⊢ (normℎ ∘ −ℎ ) = (normℎ ∘ −ℎ ) | |
17 | 15, 16 | hhims 30114 | . . . . 5 ⊢ (normℎ ∘ −ℎ ) = (IndMet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
18 | 15, 17 | hhxmet 30117 | . . . 4 ⊢ (normℎ ∘ −ℎ ) ∈ (∞Met‘ ℋ) |
19 | eqid 2736 | . . . . 5 ⊢ (MetOpen‘(normℎ ∘ −ℎ )) = (MetOpen‘(normℎ ∘ −ℎ )) | |
20 | 19 | mopntopon 23792 | . . . 4 ⊢ ((normℎ ∘ −ℎ ) ∈ (∞Met‘ ℋ) → (MetOpen‘(normℎ ∘ −ℎ )) ∈ (TopOn‘ ℋ)) |
21 | 18, 20 | mp1i 13 | . . 3 ⊢ (𝜑 → (MetOpen‘(normℎ ∘ −ℎ )) ∈ (TopOn‘ ℋ)) |
22 | hlimadd.5 | . . . 4 ⊢ (𝜑 → 𝐹 ⇝𝑣 𝐴) | |
23 | 15 | hhnv 30107 | . . . . . . 7 ⊢ 〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ NrmCVec |
24 | df-hba 29911 | . . . . . . 7 ⊢ ℋ = (BaseSet‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) | |
25 | 15, 23, 24, 17, 19 | h2hlm 29922 | . . . . . 6 ⊢ ⇝𝑣 = ((⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) ↾ ( ℋ ↑m ℕ)) |
26 | resss 5962 | . . . . . 6 ⊢ ((⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) ↾ ( ℋ ↑m ℕ)) ⊆ (⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) | |
27 | 25, 26 | eqsstri 3978 | . . . . 5 ⊢ ⇝𝑣 ⊆ (⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) |
28 | 27 | ssbri 5150 | . . . 4 ⊢ (𝐹 ⇝𝑣 𝐴 → 𝐹(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))𝐴) |
29 | 22, 28 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))𝐴) |
30 | hlimadd.6 | . . . 4 ⊢ (𝜑 → 𝐺 ⇝𝑣 𝐵) | |
31 | 27 | ssbri 5150 | . . . 4 ⊢ (𝐺 ⇝𝑣 𝐵 → 𝐺(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))𝐵) |
32 | 30, 31 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))𝐵) |
33 | 15 | hhva 30108 | . . . . 5 ⊢ +ℎ = ( +𝑣 ‘〈〈 +ℎ , ·ℎ 〉, normℎ〉) |
34 | 17, 19, 33 | vacn 29636 | . . . 4 ⊢ (〈〈 +ℎ , ·ℎ 〉, normℎ〉 ∈ NrmCVec → +ℎ ∈ (((MetOpen‘(normℎ ∘ −ℎ )) ×t (MetOpen‘(normℎ ∘ −ℎ ))) Cn (MetOpen‘(normℎ ∘ −ℎ )))) |
35 | 23, 34 | mp1i 13 | . . 3 ⊢ (𝜑 → +ℎ ∈ (((MetOpen‘(normℎ ∘ −ℎ )) ×t (MetOpen‘(normℎ ∘ −ℎ ))) Cn (MetOpen‘(normℎ ∘ −ℎ )))) |
36 | 13, 14, 21, 21, 1, 3, 29, 32, 35, 7 | lmcn2 23000 | . 2 ⊢ (𝜑 → 𝐻(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))(𝐴 +ℎ 𝐵)) |
37 | 25 | breqi 5111 | . . 3 ⊢ (𝐻 ⇝𝑣 (𝐴 +ℎ 𝐵) ↔ 𝐻((⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) ↾ ( ℋ ↑m ℕ))(𝐴 +ℎ 𝐵)) |
38 | ovex 7390 | . . . 4 ⊢ (𝐴 +ℎ 𝐵) ∈ V | |
39 | 38 | brresi 5946 | . . 3 ⊢ (𝐻((⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ ))) ↾ ( ℋ ↑m ℕ))(𝐴 +ℎ 𝐵) ↔ (𝐻 ∈ ( ℋ ↑m ℕ) ∧ 𝐻(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))(𝐴 +ℎ 𝐵))) |
40 | 37, 39 | bitri 274 | . 2 ⊢ (𝐻 ⇝𝑣 (𝐴 +ℎ 𝐵) ↔ (𝐻 ∈ ( ℋ ↑m ℕ) ∧ 𝐻(⇝𝑡‘(MetOpen‘(normℎ ∘ −ℎ )))(𝐴 +ℎ 𝐵))) |
41 | 12, 36, 40 | sylanbrc 583 | 1 ⊢ (𝜑 → 𝐻 ⇝𝑣 (𝐴 +ℎ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 〈cop 4592 class class class wbr 5105 ↦ cmpt 5188 ↾ cres 5635 ∘ ccom 5637 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 ↑m cmap 8765 1c1 11052 ℕcn 12153 ∞Metcxmet 20781 MetOpencmopn 20786 TopOnctopon 22259 Cn ccn 22575 ⇝𝑡clm 22577 ×t ctx 22911 NrmCVeccnv 29526 ℋchba 29861 +ℎ cva 29862 ·ℎ csm 29863 normℎcno 29865 −ℎ cmv 29867 ⇝𝑣 chli 29869 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128 ax-pre-sup 11129 ax-addf 11130 ax-mulf 11131 ax-hilex 29941 ax-hfvadd 29942 ax-hvcom 29943 ax-hvass 29944 ax-hv0cl 29945 ax-hvaddid 29946 ax-hfvmul 29947 ax-hvmulid 29948 ax-hvmulass 29949 ax-hvdistr1 29950 ax-hvdistr2 29951 ax-hvmul0 29952 ax-hfi 30021 ax-his1 30024 ax-his2 30025 ax-his3 30026 ax-his4 30027 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-tp 4591 df-op 4593 df-uni 4866 df-int 4908 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-se 5589 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-isom 6505 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-of 7617 df-om 7803 df-1st 7921 df-2nd 7922 df-supp 8093 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-1o 8412 df-2o 8413 df-er 8648 df-map 8767 df-pm 8768 df-ixp 8836 df-en 8884 df-dom 8885 df-sdom 8886 df-fin 8887 df-fsupp 9306 df-fi 9347 df-sup 9378 df-inf 9379 df-oi 9446 df-card 9875 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-div 11813 df-nn 12154 df-2 12216 df-3 12217 df-4 12218 df-5 12219 df-6 12220 df-7 12221 df-8 12222 df-9 12223 df-n0 12414 df-z 12500 df-dec 12619 df-uz 12764 df-q 12874 df-rp 12916 df-xneg 13033 df-xadd 13034 df-xmul 13035 df-icc 13271 df-fz 13425 df-fzo 13568 df-seq 13907 df-exp 13968 df-hash 14231 df-cj 14984 df-re 14985 df-im 14986 df-sqrt 15120 df-abs 15121 df-struct 17019 df-sets 17036 df-slot 17054 df-ndx 17066 df-base 17084 df-ress 17113 df-plusg 17146 df-mulr 17147 df-sca 17149 df-vsca 17150 df-ip 17151 df-tset 17152 df-ple 17153 df-ds 17155 df-hom 17157 df-cco 17158 df-rest 17304 df-topn 17305 df-0g 17323 df-gsum 17324 df-topgen 17325 df-pt 17326 df-prds 17329 df-xrs 17384 df-qtop 17389 df-imas 17390 df-xps 17392 df-mre 17466 df-mrc 17467 df-acs 17469 df-mgm 18497 df-sgrp 18546 df-mnd 18557 df-submnd 18602 df-mulg 18873 df-cntz 19097 df-cmn 19564 df-psmet 20788 df-xmet 20789 df-met 20790 df-bl 20791 df-mopn 20792 df-top 22243 df-topon 22260 df-topsp 22282 df-bases 22296 df-cn 22578 df-cnp 22579 df-lm 22580 df-tx 22913 df-hmeo 23106 df-xms 23673 df-tms 23675 df-grpo 29435 df-gid 29436 df-ginv 29437 df-gdiv 29438 df-ablo 29487 df-vc 29501 df-nv 29534 df-va 29537 df-ba 29538 df-sm 29539 df-0v 29540 df-vs 29541 df-nmcv 29542 df-ims 29543 df-hnorm 29910 df-hba 29911 df-hvsub 29913 df-hlim 29914 |
This theorem is referenced by: chscllem4 30582 |
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